There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article (...) lays out and compares these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

This chapter argues that the standard conception of Spinoza as a fellow-travelling mechanical philosopher and proto-scientific naturalist is misleading. It argues, first, that Spinoza’s account of the proper method for the study of nature presented in the Theological-Political Treatise (TTP) points away from the one commonly associated with the mechanical philosophy. Moreover, throughout his works Spinoza’s views on the very possibility of knowledge of nature are decidedly sceptical (as specified below). Third, in the seventeenth-century debates over proper methods in (...) the sciences, Spinoza sided with those that criticized the aspirations of those (the physico-mathematicians, Galileo, Huygens, Wallis, Wren, etc) who thought the application of mathematics to nature was the way to make progress. In particular, he offers grounds for doubting their confidence in the significance of measurement as well as their piece-meal methodology (see section 2). Along the way, this chapter offers a new interpretation of common notions in the context of treating Spinoza’s account of motion (see section 3). (shrink)

In this paper, I argue that a naturalist approach in philosophy of mathematics justifies a pluralist conception of set theory. For the pluralist, there is not a Single Universe, but there is rather a Multiverse, composed by a plurality of universes generated by various set theories. In order to justify a pluralistic approach to sets, I apply the two naturalistic principles developed by Penelope Maddy (cfr. Maddy (1997)), UNIFY and MAXIMIZE, and analyze through them the potential of the (...) set theoretic multiverse to be the best framework for mathematical practice. According to UNIFY, an adequate set theory should be foundational, in the sense that it should allow one to represent all the currently accepted mathematical theories. As for MAXIMIZE, this states that any adequate set theory should be as powerful as possible, allowing one to prove as many results and isomorphisms as possible. In a recent paper, Maddy (2017) has argued that this two principle justify ZFC as the best framework for mathematical practice. I argue that, pace Maddy, these two principles justify a multiverse conception of set theory, more precisely, the generic multiverse with a core (GMH). (shrink)

Throughout his work, John Dewey seeks to emancipate philosophical reflection from the influence of the classical tradition he traces back to Plato and Aristotle. For Dewey, this tradition rests upon a conception of knowledge based on the separation between theory and practice, which is incompatible with the structure of scientific inquiry. Philosophical work can make progress only if it is freed from its traditional heritage, i.e. only if it undergoes reconstruction. In this study I show that implicit appeals to the (...) classical tradition shape prominent debates in philosophy of mathematics, and I initiate a project of reconstruction within this field. (shrink)

What is the ontological status of mathematical structures? Michael Resnic, Stewart Shapiro and Gianluigi Oliveri, are contemporaries of American philosophers on mathematics, they give Platonic answers on this question.

ABSTRACT. May scientists rely on substantive, a priori presuppositions? Quinean naturalists say "no," but Michael Friedman and others claim that such a view cannot be squared with the actual history of science. To make his case, Friedman offers Newton's universal law of gravitation and Einstein's theory of relativity as examples of admired theories that both employ presuppositions (usually of a mathematical nature), presuppositions that do not face empirical evidence directly. In fact, Friedman claims that the use of such presuppositions is (...) a hallmark of "science as we know it." But what should we say about the special sciences, which typically do not rely on the abstruse formalisms one finds in the exact sciences? I identify a type of a priori presupposition that plays an especially striking role in the development of empirical psychology. These are ontological presuppositions about the type of object a given science purports to study. I show how such presuppositions can be both a priori and rational by investigating their role in an early flap over psychology's contested status as a natural science. The flap focused on one of the field's earliest textbooks, William James's Principles of Psychology. The work was attacked precisely for its reliance on a priori presuppositions about what James had called the "mental state," psychology's (alleged) proper object. I argue that the specific presuppositions James packed into his definition of the "mental state" were not directly responsible to empirical evidence, and so in that sense were a priori; but the presuppositions were rational in that they were crafted to help overcome philosophical objections (championed by neo-Hegelians) to the very idea that there can be a genuine science of mind. Thus, my case study gives an example of substantive, a priori presuppositions being put to use—to rational use—in the special sciences. In addition to evaluating James's use of presuppositions, my paper also offers historical reflections on two different strands of pragmatist philosophy of science. One strand, tracing back through Quine to C. S. Peirce, is more naturalistic, eschewing the use of a priori elements in science. The other strand, tracing back through Kuhn and C. I. Lewis to James, is more friendly to such presuppositions, and to that extent bears affinity with the positivist tradition Friedman occupies. (shrink)

The paper is the final, fifth part of a series of studies introducing the new conceptions of “Hilbert mathematics” and “ontomathematics”. The specific subject of the present investigation is the proper philosophical sense of both, including philosophy of mathematics and philosophy of physics not less than the traditional “first philosophy” (as far as ontomathematics is a conservative generalization of ontology as well as of Heidegger’s “fundamental ontology” though in a sense) and history of philosophy (...) (deepening Heidegger’s destruction of it from the pre-Socratics to the Pythagoreans). Husserl’s phenomenology and Heidegger’s derivative “fundamental ontology” as well as his later doctrine after the “turn” are the starting point of the research as established and well known approaches relative to the newly introduced conception of ontomathematics, even more so that Husserl himself started criticizing his “Philosophy of arithmetic” as too naturalistic and psychological turning to “Logical investigations” and the foundations of phenomenology. Heidegger’s “Aletheia” is also interpreted ontomathematically: as a relation of locality and nonlocality, respectively as a motion from nonlocality to locality if both are physically considered. Aristotle’s ontological revision of Plato’s doctrine is “destructed” further from the pre-Socratics' “Logos” or Heideger’s “Language” (after the “turn”) to the Pythagoreans “Numbers” or “Arithmetics” as an inherent and fundamental philosophical doctrine. Then, a leap to contemporary physics elucidates the essence of ontomathematics overcoming the Cartesian abyss inherited from Plato’s opposition of “ideas” versus “things”, and now unifying physics and mathematics, particularly allowing for the “creation from nothing” instead of the quasi-scientific myth of the “Big Bang”. Furthermore, ontomathematics needs another interpretation of arithmetic, propositional logic and set theory in the foundations of mathematics, where the latter two ones are both identified with Boolean algebra, and the former is considered to be a “half of Boolean algebra” in the exact meaning to be equated to it after doubling by a dual anti-isometric counterpart of Peano arithmetic. That unified algebraic realization of the foundations of mathematics is related to Hilbert mathematics in both “narrow and wide senses” where the latter is isomorphic to the qubit Hilbert space, thus underlying all the physical world by the newly introduced substance of quantum information being physically dimensionless and generalizing classical information measured in bits. The substance of information, whether classical or quantum, visualizes the way of the unification of physics and mathematics by merging their foundations in Hilbert arithmetic and Hilbert mathematics: thus how ontomathetics is a “first philosophy”. The relation of ontomathematics to the Socratic “human problematics”, furthermore being fundamental for Western philosophy in Modernity, is discussed. Ontomathematics implies its “substitution” by abstract information (or by “subjectless choice” relevant to it), thus “obliterating the human outline on the ocean beach sand” (by Michel Foucault’s metaphor). A reflection back from the viewpoint of mathematic to Western philosophy as the philosophy of locality ends the study. (shrink)

The Polish philosophy of mathematics in the 19th century had its origins in the Romantic period under the influence of the then-predominant idealist philosophies. The decline of Romantic philosophy precipitated changes in general philosophy, but what is less well known is how it triggered changes in the philosophy of mathematics. In this paper, we discuss how the Polish philosophy of mathematics evolved from the metaphysical approach that had been formed during the Romantic (...) era to the more modern positivistic paradigm. These evolutionary changes are attributed to the philosophers Henryk Struve, Antoni Molicki and Julian Ochorowicz, and mathematicians Karol Hertz and Samuel Dickstein. We also show how implicit ideas (i.e., those not declared openly) from the area between the philosophy of science and general philosophy played a crucial role in the paradigm shift in the Polish philosophy of mathematics. (shrink)

I claim that a relatively new position in philosophy of mathematics, pluralism, overlaps in striking ways with the much older Jain doctrine of anekantavada and the associated doctrines of nyayavada and syadvada. I first outline the pluralist position, following this with a sketch of the Jain doctrine of anekantavada. I then note the srrong points of overlaps and the morals of this comparison of pluralism and anekantavada.

It is undeniable Poincaré was a very famous and influential scientist. So, possibly because of it, it was relatively easy for him to participate in the heated discussions of the foundations of mathematics in the early 20th century. We can say it was “easy” because he didn't get involved in this subject by writing great treatises, or entire books about his own philosophy of mathematics (as other authors from the same period did). Poincaré contributed to the (...) class='Hi'>philosophy of mathematics by writing short essays and letters. (shrink)

Any philosophy of science ought to have something to say about the nature of mathematics, especially an account like constructive empiricism in which mathematical concepts like model and isomorphism play a central role. This thesis is a contribution to the larger project of formulating a constructive empiricist account of mathematics. The philosophy of mathematics developed is fictionalist, with an anti-realist metaphysics. In the thesis, van Fraassen's constructive empiricism is defended and various accounts of mathematics (...) are considered and rejected. Constructive empiricism cannot be realist about abstract objects; it must reject even the realism advocated by otherwise ontologically restrained and epistemologically empiricist indispensability theorists. Indispensability arguments rely on the kind of inference to the best explanation the rejection of which is definitive of constructive empiricism. On the other hand, formalist and logicist anti-realist positions are also shown to be untenable. It is argued that a constructive empiricist philosophy of mathematics must be fictionalist. Borrowing and developing elements from both Philip Kitcher's constructive naturalism and Kendall Walton's theory of fiction, the account of mathematics advanced treats mathematics as a collection of stories told about an ideal agent and mathematical objects as fictions. The account explains what true portions of mathematics are about and why mathematics is useful, even while it is a story about an ideal agent operating in an ideal world; it connects theory and practice in mathematics with human experience of the phenomenal world. At the same time, the make-believe and game-playing aspects of the theory show how we can make sense of mathematics as fiction, as stories, without either undermining that explanation or being forced to accept abstract mathematical objects into our ontology. All of this occurs within the framework that constructive empiricism itself provides the epistemological limitations it mandates, the semantic view of theories, and an emphasis on the pragmatic dimension of our theories, our explanations, and of our relation to the language we use. (shrink)

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, (...) are parts of the physical world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio (...) of two heights, for example, is a perceivable and measurable real relation between properties of physical things, a relation that can be shared by the ratio of two weights or two time intervals. Ratios are an example of continuous quantity; discrete quantities, such as whole numbers, are also realised as relations between a heap and a unit-making universal. For example, the relation between foliage and being-a-leaf is the number of leaves on a tree,a relation that may equal the relation between a heap of shoes and being-a-shoe. Modern higher mathematics, however, deals with some real properties that are not naturally seen as quantity, so that the “science of quantity” theory of mathematics needs supplementation. Symmetry, topology and similar structural properties are studied by mathematics, but are about pattern, structure or arrangement rather than quantity. (shrink)

Abstract In the new millennium there have been important empirical developments in the philosophy of mathematics. One of these is the so-called “Empirical Philosophy of Mathematics”(EPM) of Buldt, Löwe, Müller and Müller-Hill, which aims to complement the methodology of the philosophy of mathematics with empirical work. Among other things, this includes surveys of mathematicians, which EPM believes to give philosophically important results. In this paper I take a critical look at the sociological part of (...) EPM as a case study of ... (shrink)

I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over mathematical realism and (...) fictionalism. (shrink)

The dynamic nature of physics cannot be captured through an exclusive focus on the static mathematical formulations of physical theories. Instead, we can more fruitfully think of physics as a set of distinctively social, cognitive, and theoretical/methodological practices. An emphasis on practice has been one of the most notable aspects of the recent “naturalistic turn” in general philosophy of science, in no small part due to the arguments of many feminist philosophers of science. A major project of feminist (...) class='Hi'>philosophy of physics has been to shine a critical light on the social and cognitive practices in physics, and how those ultimately influence other aspects of the science. Here we argue that traditional philosophy of physics has focused exclusively on the theoretical/methodological practices of physics, and that feminist philosophy of physics seeks to broaden the focus to include the social and cognitive practices as well. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

What is the role of mathematics in scientific explanations? Does it/can it play an explanatory part? This question is at the core of the recent ontological debate in the philosophy of mathematics. My aim in this paper is to argue that the two main approaches to this problem found in recent literature (i.e. the top-down and the bottom-up approaches) are both deeply problematic. This has an important implication for the dispute over the existence of mathematical entities: to (...) make progress possible in this debate, we either have to find a different approach to the problem of the role of mathematics in scientific explanations (one that is not affected by the problems that, as I argue in this paper, plague the existing approaches), or we need to recast it in different terms. (shrink)

Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras’ Theorem or Fermat’s Last Theorem? The metaphysical question of what numbers are and (...) the epistemological question of how we know about them are central to the philosophy of mathematics. These and related philosophical questions are of particular interest because of mathematics’ unusual status. Mathematics is exceptional in that, on the one hand, it appears unhesitatingly true—no one doubts that 2 + 3 = 5—but on the other, as just noted, it is not about the physical world. This ambivalent status is what gives the philosophy of mathematics its special interest. The philosophy of mathematics is also one of the oldest academic fields, more or less coeval with philosophy itself. But contemporary philosophy of mathematics is rather different from its pre-twentieth-century antecedents, largely for three reasons. The first is that since the seventeenth century, mathematics has become integral to science. Science has over the past few centuries become increasingly mathematical, and indeed the fundamental science of nature, physics, is today recognised as a branch of applied mathematics. The second is that mathematics underwent a transformation in the course of nineteenth century: having started the century as a rather traditional-looking science of quantity it emerged a hundred years later a radically transformed abstract theory of structure. The final factor in the transformation of the philosophy of mathematics is the rise of modern logic. Developed by Frege, Cantor and others in the late nineteenth century, modern logic pervades contemporary mathematics, philosophy and computer science, and has had an immeasurable effect on the philosophy of mathematics. These volumes will collect the major works in this major field, with a focus on the last few decades. The anthology will include technical work, which interprets philosophically significant mathematical results or subfields of mathematics, as well as purely philosophical writing, aimed at those without advanced mathematics. The collection should be of interest to both philosophers and mathematicians, as well as to anyone who is susceptible to wondering what the main intellectual tool used in science, economics and finance, and indeed everyday life is ultimately about. (shrink)

Since the publication of the Remarks on the Foundations of Mathematics, Wittgenstein’s interpreters have endeavored to reconcile his general constructivist/anti-realist attitude towards mathematics with his confessed anti-revisionary philosophy. In this article, the author revisits the issue and presents a solution. The basic idea consists in exploring the fact that the so-called “non-constructive results” could be interpreted so that they do not appear non-constructive at all. The author substantiates this solution by showing how the translation of mathematical results, (...) given by the possibility of translation between logics, can be seen as a tool for partially implementing the solution Wittgenstein had in mind. (shrink)

As analytic philosophy is becoming increasingly aware of and interested in its own history, the study of that field is broadening to include, not just its earliest beginnings, but also the mid-twentieth century. One of the towering figures of this epoch is W.V. Quine (1908-2000), champion of naturalism in philosophy of science, pioneer of mathematical logic, trying to unite an austerely physicalist theory of the world with the truths of mathematics, psychology, and linguistics. Quine's posthumous papers, (...) notes, and drafts revealing the development of his views in the forties have recently begun to be published, as well as careful philosophical studies of, for instance, the evolution of his key doctrine that mathematical and logical truth are continuous with, not divorced from, the truths of natural science. But one central text has remained unexplored: Quine's Portuguese-language book on logic, his 'farewell for now' to the discipline as he embarked on an assignment in the Navy in WWII. Anglophone philosophers have neglected this book because they could not read it. Jointly with colleagues, I have completed the first full English translation of this book. In this accompanying paper I draw out the main philosophical contributions Quine made in the book, placing them in their historical context and relating them to Quine's overall philosophical development during the period. Besides significant developments in the evolution of Quine's views on meaning and analyticity, I argue, this book is also driven by Quine's indebtedness to Russell and Whitehead, Tarski, and Frege, and contains crucial developments in his thinking on philosophy of logic and ontology. This includes early versions of some arguments from 'On What There Is', four-dimensionalism, and virtual set theory. (shrink)

Indonesia has recently faced problems in various aspects of life. The results of a social media survey in Indonesia in early 2021 that the biggest threat to the Pancasila ideology is communism and other western ideologies. Communism has a dark history in the life of the Indonesian people. It shows the problem of thinking and philosophical views of the Indonesian people. This research is textbook research that aims to analyze philosophy books, namely mathematics education philosophy textbooks written (...) with a Western cultural background by Paul Ernest. This book was chosen as the object of analysis because this book is the essential reference textbook for the philosophy of mathematics education in universities providing prospective mathematics teachers in Indonesia. The research results show that this book contributes to developing Western thoughts that are contrary to Pancasila. It needs to eliminate the most critical thing in this book. It is the atheistic and individualistic philosophy underlying the ideology of mathematics education because this is not in line with the theistic collectivistic Pancasila. Therefore, this research recommends developing a textbook on the philosophy of mathematics education. It should be under the Pancasila ideology or written based on the Pancasila ideology. (shrink)

Imre Lakatos' views on the philosophy of mathematics are important and they have often been underappreciated. The most obvious lacuna in this respect is the lack of detailed discussion and analysis of his 1976a paper and its implications for the methodology of mathematics, particularly its implications with respect to argumentation and the matter of how truths are established in mathematics. The most important themes that run through his work on the philosophy of mathematics and (...) which culminate in the 1976a paper are (1) the (quasi-)empirical character of mathematics and (2) the rejection of axiomatic deductivism as the basis of mathematical knowledge. In this paper Lakatos' later views on the quasi-empirical nature of mathematical theories and methodology are examined and specific attention is paid to what this view implies about the nature of mathematical argumentation and its relation to the empirical sciences. (shrink)

A theory of mind is provided by assuming thoughts are mathematical objects (more specifically, constructible using set-theory). Problems from the philosophy of mind are probed using mathematical analogy, and the relation of minds to bodies is clarified using relations that are typical between mathematical structures.

Scientists cannot devise theories, construct models, propose explanations, make predictions, or even carry out observations, without first classifying their subject matter. The goal of scientific taxonomy is to come up with classification schemes that conform to nature's own. Another way of putting this is that science aims to devise categories that correspond to 'natural kinds.' The interest in ascertaining the real kinds of things in nature is as old as philosophy itself, but it takes on a different guise when (...) one adopts a naturalist stance in philosophy, that is when one looks closely at scientific practice and takes it as a guide for identifying natural kinds and investigating their general features. This Element surveys existing philosophical accounts of natural kinds, defends a naturalist alternative, and applies it to case studies in a diverse set of sciences. This title is also available as Open Access on Cambridge Core. (shrink)

The Polish philosophy of mathematics in the 19th century is not a well-researched topic. For this period, only five philosophers are usually mentioned, namely Jan Śniadecki, Józef Maria Hoene-Wroński, Henryk Struve, Samuel Dickstein, and Edward Stamm. This limited and incomplete perspective does not allow us to develop a well-balanced picture of the Polish philosophy of mathematics and gauge its influence on 19th- and 20th-century Polish philosophy in general. To somewhat complete our picture of the history (...) of the Polish philosophy of mathematics in those times, we here present the profiles of some lesser-known Polish Romantic philosophers of the 19th century, namely Karol Libelt, Bronisław Trentowski, and Józef Kremer. We discuss their contributions to the philosophy of mathematics and their metaphysical perspectives, and we also show how their metaphysical ideas have found some continuity in the studies of some Catholic philosophers. (shrink)

This essay offers a strategic reinterpretation of Kant’s philosophy of mathematics in Critique of Pure Reason via a broad, empirically based reconception of Kant’s conception of drawing. It begins with a general overview of Kant’s philosophy of mathematics, observing how he differentiates mathematics in the Critique from both the dynamical and the philosophical. Second, it examines how a recent wave of critical analyses of Kant’s constructivism takes up these issues, largely inspired by Hintikka’s unorthodox conception (...) of Kantian intuition. Third, it offers further analyses of three Kantian concepts vitally linked to that of drawing. It concludes with an etymologically based exploration of the seven clusters of meanings of the word drawing to gesture toward new possibilities for interpreting a Kantian philosophy of mathematics. (shrink)

The relationship between the One and the Multiple in mystic philosophy is a profound and central theme that explores the nature of existence, the cosmos, and the divine. This theme is present in various mystical traditions, including those of the East and West, and it addresses the paradoxical coexistence of the unity and multiplicity of all things. -/- In mystic philosophy, the **One** often represents the ultimate reality, the source from which all things emanate and to which all (...) things return. It is the absolute, the infinite, and the unchanging. The One is beyond all attributes and is often associated with the divine or the absolute truth. -/- The **Multiple**, on the other hand, represents the manifest world, the diversity of forms, and the realm of change and plurality. It is the world we experience through our senses, the domain of time and space, where differentiation and individuality are apparent. -/- The relationship between the One and the Multiple is not one of opposition but of emanation and unity. The Multiple is seen as a reflection, expression, or manifestation of the One. In this sense, the diversity of the world doesn’t contradict the unity of the One but rather demonstrates it in a myriad of forms. -/- Mystics seek to understand and experience this relationship through various practices and insights. They aim to transcend the illusion of separation and duality to experience the non-dual reality where the One and the Multiple are recognized as inseparable. -/- This concept can be illustrated by the metaphor of the ocean and its waves. The ocean represents the One—vast, deep, and all-encompassing—while the waves represent the Multiple—distinct, diverse, and ever-changing. Each wave is unique, yet it is not separate from the ocean. The wave’s existence is dependent on and made of the same substance as the ocean. In the same way, each individual entity in the Multiple is an expression of the One. -/- In summary, the relationship between the One and the Multiple in mystic philosophy is a dynamic interplay that challenges the conventional understanding of separation. It invites a deeper exploration of reality, where the apparent multiplicity of the world is a direct expression of the singular, underlying essence of all that is. -/- The relationship between the One and the Multiple in the context of mathematical philosophy is a profound topic that touches upon the very foundations of existence and knowledge. It’s a theme that has been explored by philosophers and mathematicians alike, often leading to the contemplation of unity and diversity within the structures of reality. -/- In mathematics, the concept of the One can be seen as the basis of unity from which all numbers derive. It’s the identity element in multiplication, the starting point in counting, and the foundation of dimension in geometry. The One is often associated with the concept of monism in philosophy, which posits that there is a single, underlying substance or principle that constitutes reality. -/- On the other hand, the Multiple represents the infinite variety and diversity of forms and numbers that arise from the One. It’s the embodiment of plurality and the complex interplay of different entities that mathematics seeks to understand and describe. This reflects the philosophical stance of pluralism, which acknowledges the existence of multiple realities or truths. -/- The interplay between the One and the Multiple can be seen in the mathematical concept of sets. A set can be thought of as a unity, a whole composed of distinct elements. Yet, each element within the set also retains its individuality, contributing to the diversity of the set’s composition. This duality is mirrored in the philosophical exploration of the universal and the particular, where the universal represents the One, and the particulars represent the Multiple. -/- In the realm of mathematical philosophy, this relationship often leads to questions about the nature of mathematical objects: Are they discovered as part of an objective reality (the One), or are they constructed by the human mind from a multitude of experiences (the Multiple)? This debate resonates with the philosophical inquiry into the nature of truth and reality. -/- Reflecting on the One and the Multiple can also lead to a deeper understanding of the self and the cosmos. Just as the number one is integral to the existence of all other numbers, the individual self can be seen as a unique expression of the universal whole. Similarly, the cosmos can be viewed as a grand unity composed of a multiplicity of forms and phenomena. -/- Gödel’s incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers. -/- Gödel’s incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers⁴. -/- The **first incompleteness theorem** reveals that within any such consistent system, there are propositions that are true but cannot be proven within the system itself. This reflects the idea of the Multiple in that there is an abundance of mathematical truths, a multiplicity that exceeds the unifying framework of any one system. It suggests that the realm of mathematical truth is more extensive than any single formal system can fully capture. -/- The **second incompleteness theorem** extends this by showing that a system cannot prove its own consistency. This relates to the concept of the One, as it implies that a system’s complete self-understanding, its unity and coherence, is unattainable from within. It must look beyond itself, to an external vantage point, to ascertain its consistency. -/- In the context of the One and the Multiple, Gödel’s theorems imply that the One (a consistent formal system) is inherently incomplete and cannot encompass the Multiple (the totality of mathematical truths). This resonates with philosophical discussions about the relationship between unity and plurality. Just as a single philosophical system cannot capture the entirety of truth, a single formal mathematical system cannot encapsulate all mathematical truths. -/- Moreover, Gödel’s work suggests that the pursuit of a single, unified theory of everything in mathematics—a One that encompasses all Multiples—is an inherently Sisyphean task. There will always be more truths (Multiples) than can be derived from any given set of axioms (the One). -/- In essence, Gödel’s incompleteness theorems provide a formal underpinning to the philosophical notion that the One cannot exist without the Multiple, and vice versa. They are interdependent, with the One giving rise to the Multiple, and the Multiple necessitating the One for its expression and comprehension. This interplay is a dance of limitations and possibilities, where the boundaries of logic, mathematics, and philosophy blur into one another. -/- The relationship between the One and the Multiple in the context of the mathematics of zero is a deeply philosophical inquiry that bridges the abstract world of numbers with the existential questions of being and non-being. -/- Zero, in mathematics, is a symbol of absence, a representation of nothingness, yet it holds a pivotal position as a number. It is the void from which all things emerge and to which they return. In the philosophy of mathematics, zero is the paradoxical junction where the One and the Multiple converge and diverge. -/- From the perspective of the One, zero can be seen as the origin—the singular point that precedes the existence of numbers. It is the empty set, the foundation upon which the edifice of mathematics is constructed. As the identity element in addition, zero maintains the integrity of numbers, for adding zero to any number leaves it unchanged, reflecting the immutable nature of the One. -/- Conversely, when we consider the Multiple, zero represents the infinite potentiality of creation. It is the canvas upon which the integers, both positive and negative, express their multitude. Zero is the balance point, the fulcrum around which the symphony of numbers dances. It embodies the plurality of possibilities, the beginning of the number line that stretches infinitely in both directions. -/- Philosophically, zero challenges our understanding of existence. It is both something and nothing—a number that quantifies the absence of quantity. This duality echoes the philosophical struggle to comprehend how the One gives rise to the Multiple. How does the unity of being manifest the diversity of the cosmos? Zero offers a mathematical metaphor for this mystery, as it encapsulates the transition from non-existence to existence, from the undifferentiated One to the differentiated Multiple. -/- In the realm of set theory, zero corresponds to the empty set—a set with no elements. This set is unique in that it is the only set that contains nothing, yet it is the foundation upon which all other sets are built. The empty set is the mathematical embodiment of the One, and all other sets, containing multiple elements, arise from it. -/- Reflecting on zero in the context of Gödel’s incompleteness theorems, we find a resonance with the idea that the One (a consistent formal system) cannot capture the entirety of the Multiple (the totality of mathematical truths). Zero, as the foundation of numbers, similarly suggests that from the nothingness of the One, the infinite complexity of the Multiple emerges—yet it can never be fully encompassed or expressed by any single system. -/- In conclusion, the mathematics of zero offers a profound reflection on the relationship between the One and the Multiple. It serves as a bridge between the abstract and the concrete, the known and the unknowable, challenging us to ponder the origins of existence and the nature of reality itself. -/- . (shrink)

This article examines the possibility of philosophizing about mathematics with children. It aims at outlining the nature of the practice of philosophy of mathematics with children in a mainly theoretical and exploratory way. First, an attempt at a definition is proposed. Second, I suggest some reasons that might motivate such a practice. My thesis is that one can identify an intrinsic as well as two extrinsic goals of philosophizing about mathematics with children. The intrinsic goal is (...) related to a presumed inherent importance of presenting children with some philosophical questions about mathematics. The extrinsic goals consist of first the positive effects such a practice can have on mathematical learning and abilities and second the fostering of children's understanding of philosophical method of inquiry and thinking and therefore of their philosophical thinking competences. Third, some examples found in the literature of previously developed ways of practising philosophy of mathematics with children are presented. This article aims at giving a general outlining picture of the issues surrounding the practice of philosophy of mathematics with children and should therefore be read as an encouragement to further development and studies. (shrink)

We give an overview of Lakatos’ life, his philosophy of mathematics and science, as well as of this issue. Firstly, we briefly delineate Lakatos’ key contributions to philosophy: his anti-formalist philosophy of mathematics, and his methodology of scientific research programmes in the philosophy of science. Secondly, we outline the themes and structure of the masterclass Lakatos’ Undone Work – The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of (...) Science, which gave rise to this special issue. Lastly, we provide a summary of the contributions to this issue. (shrink)

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a (...) formal science, mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (shrink)

This paper surveys Bolzano's Beyträge zu einer begründeteren Darstellung der Mathematik (Contributions to a better-grounded presentation of mathematics) on the 200th anniversary of its publication. The first and only published issue presents a definition of mathematics, a classification of its subdisciplines, and an essay on mathematical method, or logic. Though underdeveloped in some areas (including,somewhat surprisingly, in logic), it is nonetheless a radically innovative work, where Bolzano presents a remarkably modern account of axiomatics and the epistemology of the (...) formal sciences. We also discuss the second, unfinished and unpublished issue, where Bolzano develops his views on universal mathematics. Here we find the beginnings of his theory of collections, for him the most fundamental of the mathematical disciplines. Though not exactly the same as the later Cantorian set theory, Bolzano's theory of collections was used in very similar ways in mathematics, notably in analysis. In retrospect, Bolzano's debut in philosophy was a remarkably successful one, though its fruits would only become generally known much later. (shrink)

Why is there Philosophy of Mathematics at all? Ian Hacking. in Metascience (2015).

Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the potential to (...) broaden our view of the specific normativity of mathematics beyond logical proof. Second, some work focuses on signs to highlight the open-ended and “nonrigorous” aspects of mathematical practice, and the role of these aspects in the historical development of mathematics. The third motivation is based on the following observation: the reason differences in signs matter in mathematics is that humans are finite agents, with cognitive and computational limitations. Accordingly, studying signs can be a way to study how human computational constraints shape the mathematics that we do, and to tackle the topic of mathematical understanding. (shrink)

Penelope Maddy’s Second Philosophy is one of the most well-known ap- proaches in recent philosophy of mathematics. She applies her second-philosophical method to analyze mathematical methodology by reconstructing historical cases in a setting of means-ends relations. However, outside of Maddy’s own work, this kind of methodological analysis has not yet been extensively used and analyzed. In the present work, we will make a first step in this direction. We develop a general framework that allows us to clarify (...) the procedure and aims of the Second Philosopher’s investigation into set-theoretic methodology; pro- vides a platform to analyze the Second Philosopher’s methods themselves; and can be applied to further questions in the philosophy of set theory. (shrink)

The Reliability Challenge to moral non-naturalism has received substantial attention recently in the literature on moral epistemology. While the popularity of this particular challenge is a recent development, the challenge has a long history, as the form of this challenge can be traced back to a skeptical challenge in the philosophy of mathematics raised by Paul Benacerraf. The current Reliability Challenge is widely regarded as the most sophisticated way to develop this skeptical line of thinking, making the (...) Reliability Challenge the strongest epistemic challenge to normative nonnaturalism. In this paper, I argue that the innovations that have occurred since Benacerraf’s statement of the challenge are misconceived and confused in a number of ways. The Reliability Challenge is not the most potent epistemic challenge to moral non-naturalism. The most potent challenge comes from the fact that there is a causal condition on knowledge – or, more precisely, a becaual condition – that non-natural moral facts cannot satisfy. (shrink)

We attribute three major insights to Hegel: first, an understanding of the real numbers as the paradigmatic kind of number ; second, a recognition that a quantitative relation has three elements, which is embedded in his conception of measure; and third, a recognition of the phenomenon of divergence of measures such as in second-order or continuous phase transitions in which correlation length diverges. For ease of exposition, we will refer to these three insights as the R First Theory, Tripartite Relations, (...) and Divergence of Measures. Given the constraints of space, we emphasize the first and the third in this paper. (shrink)

In this chapter, we examine the naturalist approach in the philosophy of memory through the lens of the simulation theory of memory. On the theory, episodic memory is a kind of constructive simulation performed by a functionally specialized neurocognitive system. Taking naturalism to be a kind of methodological stance characterized by a cluster of epistemic guidelines, we illustrate the roles these guidelines have played in the development of the theory. We show how scientific evidence has guided both the (...) selection of the research question and the initial elaboration of the theory. We demonstrate how such evidence has led the simulationist to reject a familiar, commonsense picture of memory. We also trace the recent radicalization of the theory, highlighting the simulationist's continual reflection on their concepts and assumptions. In the spirit of such reflection, we end by outlining some outstanding challenges for the theory. (shrink)

I begin by outlining Du Châtelet’s ontology of mathematical objects: she is an idealist, and mathematical objects are fictions dependent on acts of abstraction. Next, I consider how this idealism can be reconciled with her endorsement of necessary truths in mathematics, which are grounded in essences that we do not create. Finally, I discuss how mathematics and physics relate within Du Châtelet’s idealism. Because the primary objects of physics are partly grounded in the same kinds of acts as (...) yield mathematical objects, she thinks we are sometimes licensed in drawing conclusions about physical things from mathematical premises. (shrink)

Nietzsche has a surprisingly significant and strikingly positive assessment of mathematics. I discuss Nietzsche's theory of the origin of mathematical practice in the division of the continuum of force, his theory of numbers, his conception of the finite and the infinite, and the relations between Nietzschean mathematics and formalism and intuitionism. I talk about the relations between math, illusion, life, and the will to truth. I distinguish life and world affirming mathematical practice from its ascetic perversion. For Nietzsche, (...) math is an artistic and moral activity that has an essential role to play in the joyful wisdom. (shrink)

This chapter considers how Liberal Naturalism interacts with the main problems and theories in the philosophy of perception. After briefly summarising the traditional philosophical problems of perception and outlining the standard philosophical theories of perceptual experience, it discusses whether a Liberal Naturalist outlook should incline one towards or away from any of these standard theories. Particular attention is paid to the work of John McDowell and Hilary Putnam, two of the most prominent Liberal Naturalists, whose work was also (...) very influential in the philosophy of perception. There is also a section focusing on colour, an especially important topic not only for debates about perceptual experience but also for debates concerning how our ‘manifest image’ of the natural world relates to our best theories in the physical sciences. (shrink)

This dissertation provides a careful reading of the later Wittgenstein’s philosophy of mathematics centered around three major themes: reality, determination, and infinity. The reading offered gives pride of place to Wittgenstein’s therapeutic conception of philosophy. This conception views questions often taken as fundamental in the philosophy of mathematics with suspicion and attempts to diagnose the confusions which lead to them. In the first essay, I explain Wittgenstein’s approach to perennial issues regarding the alleged reality to (...) which mathematical truths or propositions correspond. Wittgenstein diagnoses exotic pictures of mathematical reality as stemming from misleading analogies formed across empirical and mathematical propositions. The second essay explains Wittgenstein’s treatment of perplexity regarding the ability of a mathematical rule to determine its applications in advance. This too is found to depend on a misleading analogy, in this case across behavioral and mathematical senses of ‘determine’. The third and final essay discusses Wittgenstein’s general critical approach to “the infinite”. Philosophical perplexity about infinity is shown to depend on the verbal imagery regularly associated with proofs and their power to take hold of one’s imagination. Wittgenstein dampens the imaginative excesses often associated with “the infinite” by offering a sober redescription of Cantor’s famous method of diagonalization. (shrink)

This rich book differs from much contemporary philosophy of mathematics in the author’s witty, down to earth style, and his extensive experience as a working mathematician. It accords with the field in focusing on whether mathematical entities are real. Franklin holds that recent discussion of this has oscillated between various forms of Platonism, and various forms of nominalism. He denies nominalism by holding that universals exist and denies Platonism by holding that they are concrete, not abstract - looking (...) to Aristotle for inspiration. (shrink)

The formalist philosophy of mathematics (in its purest, most extreme version) is widely regarded as a “discredited position”. This pure and extreme version of formalism is called by some authors “game formalism”, because it is alleged to represent mathematics as a meaningless game with strings of symbols. Nevertheless, I would like to draw attention to some arguments in favour of game formalism as an appropriate philosophy of real mathematics. For the most part, these arguments have (...) not yet been used or were neglected in past discussions. (shrink)

Two slogans define structuralism: contemporary mathematics studies structures; mathematical objects are places in those structures. Shapiro’s version of structuralism posits abstract objects of three sorts. A system is “a collection of objects with certain relations” between these objects. “An extended family is a system of people with blood and marital relationships.” A baseball defense, e.g., the Yankee’s defense in the first game of the 1999 World Series, is a also a system, “a collection of people with on-field spatial and (...) ‘defensive-role’ relations”. “A structure is the abstract form of a system” ; it consists of “a collection of places [sometimes called positions or offices] and a finite collection of functions and relations on these places”. (shrink)

Hornsby is a defender of a position in the philosophy of mind she calls “naïve naturalism”. She argues that current discussions of the mind-body problem have been informed by an overly scientistic view of nature and a futile attempt by scientific naturalists to see mental processes as part of the physical universe. In her view, if naïve naturalism were adopted, the mind-body problem would disappear. I argue that her brand of anti-physicalist naturalism runs into difficulties with (...) the problem of mental causation and the completeness of physics. (shrink)

In this review, as well as discussing the pedagogical of this text book, I also discuss Linnebo's approach to the Caesar problem and the use of metaphysical notions to explicate mathematics.

The philosophy of information (PI) is a new area of research with its own field of investigation and methodology. This article, based on the Herbert A. Simon Lecture of Computing and Philosophy I gave at Carnegie Mellon University in 2001, analyses the eighteen principal open problems in PI. Section 1 introduces the analysis by outlining Herbert Simon's approach to PI. Section 2 discusses some methodological considerations about what counts as a good philosophical problem. The discussion centers on Hilbert's (...) famous analysis of the central problems in mathematics. The rest of the article is devoted to the eighteen problems. These are organized into five sections: problems in the analysis of the concept of information, in semantics, in the study of intelligence, in the relation between information and nature, and in the investigation of values. (shrink)

The essays gathered in this issue of the journal Noctua focus on the various relationships that were established between philosophy and mathematics from Galileo and Descartes to Kant, passing by Newton.